Related papers: The Higman--Thompson groups $V_n$ are $(2,2,2)$-ge…

The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these…

Let $N_{g,n}$ denote the closed non-orientable surface of genus $g$ with $n$ punctures and let ${\mathcal N}_{g,n}$ denote the mapping class group of $N_{g,n}$. Szepietowski showed that ${\mathcal N}_{g,n}$ is generated by finitely many…

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs…

The symmetric group is generated by {\sigma} = (1 2 ... n) and {\tau} = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n.

The Ewens sampling formula with parameter $\alpha$ is the distribution on $S_n$ which gives each $\pi\in S_n$ weight proportional to $\alpha^{C(\pi)}$, where $C(\pi)$ is the number of cycles of $\pi$. We show that, for any fixed $\alpha$,…

A subset $\left\{x_{1},x_{2},\hdots,x_{d}\right\}$ of a group $G$ \emph{invariably generates} $G$ if $\left\{x_{1}^{g_{1}},x_{2}^{g_{2}},\hdots,x_{d}^{g_{d}}\right\}$ generates $G$ for every $d$-tuple $(g_{1},g_{2}\hdots,g_{d})\in G^{d}$.…

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We…

Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically…

The extended mapping class group of a surface $\Sigma$ is defined to be the group of isotopy classes of (not necessarily orientation-preserving) homeomorphisms of $\Sigma$. We are able to show that the extended mapping class group of an…

We study the $S_3$-orbifold of a rank three Heisenberg vertex algebras in terms of generators and relations. By using invariant theory we prove that the orbifold algebra has a minimal strongly generating set of vectors whose conformal…

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we…

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}^{\pm}(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that $\text{Mod}^{\pm}(S_g)$ can be generated by two torsion elements.…

Let f be a Z/2Z-spin structureon a closed surface S of genus g>3. We determine a generating set of the stabilizer of f in the mapping class group of S consisting of Dehn twists about an explicit collection of 2g+1 curves in S. If g=3 then…

We construct a family of groups from suitable higher rank graphs which are analogues of the finite symmetric groups. We introduce homological invariants showing that many of our groups are, for example, not isomorphic to $nV$, when $n \geq…

We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of…

We obtain a minimal generating set of involutions for the level 2 subgroup of the mapping class group of a closed nonorientable surface.

With an arbitrary finite graph having a special form of 2-intervals (a diamond-shaped graph) we associate a subgroup of a symmetric group and a representation of this subgroup; state a series of problems on such groups and their…

Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $\Sigma_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g)$ consisting of elements…

Let $V$ be a $2n$-dimensional vector space over a field $F$ and $\Omega$ be a non-degenerate symplectic form on $V$. Denote by ${\mathfrak H}_{k}(\Omega)$ the set of all $2k$-dimensional subspaces $U\subset V$ such that the restriction…

We give a direct proof that all Higman-Thompson groups of the form $G_{k,1}$ (for $k \ge 2$) are embedded in one another, which is a recent result of N. Matte Bon. This extends the embeddings given by Higman in 1974.